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7th International Conference

„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021

June 24-26. 2021 Sarajevo, Bosnia and Herzegovina

ROLLING BALL SCULPTURE AS A MECHANICAL DESIGN

CHALLENGE

Alma Žiga1, Derzija Begic-Hajdarevic2

1(University of Zenica, Faculty of Mechanical Engineering,

aziga@mf.unze.ba)

2(University of Sarajevo, Faculty of Mechanical Engineering,

begic@mef.unsa.ba)

-----------------------------------------------------------------------------------------------------------

ABSTRACT:

Rolling ball sculpture, even the simple one, can be viewed as mechanical design challenge.

If sculpture is made of poplar plywood, then bending and twisting of track causes stresses

which can destroy rails of track. Another aspect is kinematic and dynamics of rolling ball.

Sections of track where the rails is closer together will cause the ball to roll faster, but the

ball is more likely to fall off the track. Centripetal force acting on the ball on spiral path

increases own intensity with square of velocity and might cause ball to fall off. All these

aspects will be analyzed in the paper.

Keywords: rolling ball sculpture, stress analysis in plywood semicircle console,

kinematic and dynamics of ball rolling on a track.

-----------------------------------------------------------------------------------------------------------------------

1. INTRODUCTION

A rolling ball sculpture (sometimes referred to as a marble run, ball run, gravitram,

kugelbahn, or rolling ball machine) is a form of kinetic art. Even though a rolling ball

sculpture can range from simple to extremely complex, it is always grounded to the simple

movement; a ball rolling on a track. When ball is positioned at the top of the track, and is

let to go, the gravity becomes the motive force. Utilization of creative track designs to

harness the energy of the rolling ball leads to amazing things that can be achieved. The

rolling ball can be used to captivate, demonstrate, and educate on many levels. From small

single track sculptures, to room-filling complex installations, each sculpture works in

harmony with both the environment and with the experiences of the viewers [1]. Adding

the laser cut parts, gears, handle, slots for lifting the balls, the simple movement has been

elevated to a work of mechanical art.

Idea for sculpture, described in this paper, was conceived during watching YouTube

channel Build Amazing Big Marble Run Machine [2]. Design was obtained in CAD

software SolidWorks (Fig. 1.). All parts of sculpture were made from poplar plywood,

4 mm thick. After design and analysis all parts were cut from plywood sheet by laser

cutter (Fig. 2.). Dimensions of Front and Back plate are 280x290 mm. Plates are 16 mm

set apart. Between them is big toothed wheel with slots for balls to drop in when rolled

down the track. Big wheel meshes with three small gears, one of which has handle to set

big wheel in motion and to lift balls to the beginning of the track. The track forms left-

hand, helix spiral with two revolutions. Spiral track has central dimensions: radius 120

mm and height 160 mm. Cross section of rail is rectangle 8x4mm. Rails is spaced for 8

mm (Fig. 7c). Balls are made of steel and have diameters of 11.4 mm. For assembly, track

was made of four, half-revolution spirals. As the two rails of the track should be bent and

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka

7th International Conference

„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021

June 24-27. 2021 Sarajevo, Bosnia and Herzegovina

twisted to obtain 40 mm deflection, stress analysis should be done for inner rail, with

smaller diameter. Another design concern is distance between rails as it means ball

stability and ratio of its translational to rotational velocity. And the last analysis is dynamic

of ball rolling on track.

Fig. 1. SolidWorks design of rolling ball sculpture: 1-Front plate, 2-Back Plate,

3-Toothed wheel with slots, 4-Small gear, 5-Handle, 6-Track and 7-Ball

Fig. 2. Rolling ball sculpture

1

2

6

3

5

4

7

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika

7th International Conference

„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021

June 24-26. 2021 Sarajevo, Bosnia and Herzegovina

2. DEFLECTION OF INNER RAIL AS A HALF-CIRCLE CONSOLE

The deflection of the half-circle console end (at the point of application of the force P and

in the direction of the force, Fig. 3) will be determined, as it is done in papers: [3], Horibe

T. & Mori K. (2015), [4], Dahlberg T. (2004) and [5], Žiga A. et al. (2018). The force P

is normal to the xy plane.

Fig. 3. Half-circle console

Fig. 4. Moments equilibrium

Fig. 4 shows cross-section of the beam situated at angle

ϕ

. The bending moment Mb and

torsional moment Mt are acting at this cross-section. The shear force has been omitted in

the figure, since its influence on beam deflection can be neglected. The equilibrium of

moments is used, equations are obtained and solving for Mb and Mt gives:

[ ]

[ ]

cos ( )sin

( )cos sin

b

t

M Py R x

M P Rx y

ϕϕ

ϕϕ

=− +−

= −+ +

(1)

Elastic strain energy stored in the beam is:

22

00

t

11

dd

22

LL

bt

U Ms Ms

EI GK

= +

∫∫

(2)

Where: L is the length of the half-circle console, E is modulus of elasticity, G is shear

modulus, I is second moment of cross-sectional area, Kt is the cross-sectional factor of

torsional rigidity.

Using Castigliano’s theorem, the deflection of the console end, at the load P, can be

calculated:

00

11

2 d 2d

22

LL

bt

bt

t

MM

UM s Ms

P EI P GK P

δ

∂∂

∂

= = +

∂∂ ∂

∫∫

(3)

With Mb, Mt,

∂

Mb/

∂

P and

∂

Mt/

∂

P from (1), deflection is:

[ ] [ ]

22

00

cos ( )sin ( )cos sidnd

LL

t

PP

y Rx s Rx y s

EI GK

δ ϕ ϕ ϕϕ

= − − − + −+ +

∫∫

(4)

Expressions for

sin ,cos ,d , ,d /dxsy y

ϕϕ

can be obtained by geometry (Fig. 3.) and

replaced in Equation (4). The integration over ds from 0 to L becomes integration over dx

x

x

y

R

y

ϕ

s

P

M

b

M

t

ϕ

dx

dy

ds

ϕ

τ

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka

7th International Conference

„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021

June 24-27. 2021 Sarajevo, Bosnia and Herzegovina

from R to -R. With substitution of dimensionless integration variable t=x/R, the integrals

have limits 1 and -1.

3 32

2

11

11

(1 )1

1d d

(1 )

t

PR PR t t

tt t

EI GK t

δ

−−

−+ −

= −− + +

∫∫

(5)

33

1,5708 4,71239

t

PR PR

EI GK

δ

= +

(6)

Deflection of half-circle console consists of two parts: due to bending and due to torsion.

So in order to evaluate contribution of these parts to whole deflection, isotropic wooden

material will be considered. Poplar veneer has three modulus of elasticity and three values

of Poisson ratio. These values have been taken from the paper [6], Brezović, Mladen

Vladimir, J. and Stjepan, P. (2003). One mean value is calculated for modulus of elasticity

and Poisson ratio: E = 3633 MPa,

ν

= 0.35. Shear modulus is calculated by the expression:

G=E / [2(1+

ν

)] and has a value: G = 1346 MPa. Dimensions of half-circle, inner track are:

radius-R=112 mm and cross-sectional dimensions (bxh) - 8x4 mm. Axial moment of

inertia is

33 4

/12 8 4 /12 42.67 mm

x

I bh=⋅=⋅=

.

The factor of torsional rigidity is

33 4

2

0,229 8 4 117.25 mm

t

K cb h= ⋅ = ⋅⋅ =

, where c2 is

coefficient for rectangular bar in torsion, taken from the book [7], Beer, F. P. (2014),

Mechanics of Materials.

Solving equation (6) for unknown force, where, at the end of half-circle console, deflection

is

δ

=40 mm, gives force value of P=0.7117 N. There, part of deflection due to bending is

10.1 mm and part of deflection due to torsion is 29.9 mm. So for isotropic wooden half-

circle console, torsion has much more influence on deflection than bending.

Coordinates of the console clamp are:

, cos 112 mm, sin 0.xR yR

ϕπ ϕ ϕ

===−==

Using equation (1), bending and torsional moments in the clamp are:

0, 159.42Nmm

bt

MM= =

.

Bending and torsional stresses in the clamp are:

22

1

60 5.063 MPa

bt

bend torsion

MM

bh c bh

στ

= = = =

(7)

Where coefficient c1 depends only upon the ratio b and h. For b/h = 8/4 = 2, coefficient is:

c1=0,246.

With stress element

σ

x = 0,

σ

y =

σ

bend =0 and

τ

x =

τ

torsion = 5.063 MPa, principal stresses

are:

2

2

1,2

5.063MPa

22

xy xy xy

σσ σσ

στ

+−

= ± +=±

(8)

Ekvivalent, von Mises stress is:

22

1 12 2 8.77MPa

vonMises

σ σ σσ σ

= − +=

(9)

For numerical analysis of deflection, the inner rail was modeled and meshed (Fig. 5). The

left side was clamped and at the right side, vertical translation of outer, lower vertex was

set to be 40 mm. The study gave maximal, von Mises stress of the value 8.791 MPa, near

the clamp (Fig. 6).

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika

7th International Conference

„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021

June 24-26. 2021 Sarajevo, Bosnia and Herzegovina

Fig. 5. Boundary conditions and

meshing

Fig. 6. Von Mises stress in inner track due to

deflection

3. TRACK RAILS SPACING AND BALL KINEMATICS

The spacing between the two rails of the track is an optimization between the security of

the ball on the track and ball spin relative to its linear velocity.

Consider a ball rolling over a horizontal, frictional surface (Fig. 7a). Let vC be the

translational velocity of the ball's center of mass, and let

ω

be the angular velocity of the

ball about an axis passing through its center of mass. Consider the point B of contact

between the ball and the surface. The velocity vB of this point is made up of two

components: the translational velocity vC, which is common to all elements of the ball,

and the tangential velocity vt =

ω⋅

R due to the ball's rotational motion. Thus, vB = vC – vt =

vC −

ω⋅

R. Suppose that the ball rolls without slipping. In other words, suppose that there is

no frictional energy dissipation as the ball moves over the surface. This is only possible if

there is zero net motion between the surface and the bottom of the ball, which implies

vB = 0 or vC = vt or vC =

ω⋅

R. The ratio of translational velocity to the tangential velocity of

the bottom of the ball is: vC / vt =1.

However, if the point that the ball is rolling on changes to two points, the ratio changes. A

certain angle θ is subtended by the radius of ball contact point with the vertical (Fig. 7c).

Let

ω

be the angular velocity of the ball. Let vC be the translational velocity of the ball's

center of mass: vC =

ω⋅

b (Fig. 7b), where b is the height of the center of mass from the axis

connecting the points of contact. Consider the point B at the bottom of the ball. The

velocity vB of this point is made up of two components: the translational velocity vC and

the tangential velocity vt =

ω⋅

R due to the ball's rotational motion. Thus, vB = vC – vt =

ω⋅

b −

ω⋅

R. The ratio of translational velocity to the tangential velocity of the bottom of the

ball is: vC / vt = b / R = cos

θ

.

Sections of track where the rails were closer together would cause the ball to roll faster, at

the cost of stability, as the ball was more likely to fall off the track. Increasing the distance

between the rails would cause the ball to roll slower, but would increase the odds that the

ball stayed on the track.

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka

7th International Conference

„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021

June 24-27. 2021 Sarajevo, Bosnia and Herzegovina

Fig. 7. A ball rolling over a rough surface: a) one point contact b) two point contact c)

distance between rails.

For the given track, distance between rails is d = 8 mm. For radius of steel ball R = 5.7

mm, angle subtended by the radius of ball contact point with the vertical is

θ

= 44.56°.

The ratio of translational velocity to tangential velocity of bottom ball point is vC / vt =

b / R = cos

θ

= 0.71. This angle yields a good balance between security on the track and

translational velocity to tangential velocity ratio.

3. BALL DYNAMICS DURING ROLLING DOWN THE TRACK

Fig. 8. A ball rolling down: a) natural coordinate system b) all forces and moments

acting on the ball

Using d'Alembert's principle, rolling ball can be transformed into an equivalent static

system by adding the so-called "inertial forces" and "inertial torques" or moments [8]. The

ball can then be analyzed as a static system subjected to this "inertial forces and moments"

and the external forces and reactions. Assuming that friction is negligible, forces that act

on rolling ball are (Fig. 8b): inertial forces due to tangential and centripetal acceleration,

0.1

0.0

0.1

x,m

0.1

0.0

0.1

y,m

0.15

0.10

0.05

0.00

z,m

θ

θ

ω

v

C

b

R

ω

= v

C

/ b

v

t / vC = cosθ

vB= vC- vt

ω

v

C

R

ω

= v

C

/ R

v

t / vC = 1

v

t

v

t

C

B

a)

v

B

= v

C

- v

t

=0

b

R

c)

d

v

t

v

t

b)

B

C

v

B

B

N

T

F

a in

mg

F

cp

in

Fo

F

i

M

in

θ

θ

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika

7th International Conference

„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021

June 24-26. 2021 Sarajevo, Bosnia and Herzegovina

inertial moment due to angular acceleration, reaction forces due to contacts with inside

and outside rail, and active force of gravity.

The local, normal and tangential (natural) coordinate system for certain points along the

path are shown in Fig. 8a, using Mathematica software [9]. For given a parameterized path

r(s), definition of unit vectors in tangential, normal and binormal directions are:

'(s)

'(s) '(s) ''(s)

'(s) '(s) '(s) ''(s)

T

TN B

T

u

r rr

uu u

r u rr

×

= = = ×

(10)

Path as a vector is

0.04

(s) 0.12cost,0.12sint,

rt

π

= −

, where t is in interval

{ }

,0, 4t

π

for

two revolutions spiral.

Static equilibrium conditions in normal and binormal coordinate system are:

0 sin sin 0

0 cos cos . 0

in

N i o i cp

Bi o i B

F FFF

F F F mg u

θθ

θθ

Σ= − + =

Σ= + + =

(11)

Fcpin is centripetal force,

2in

cp

F mv

ν

=

, where

ν

is a curve of the path:

3

'(s) ''(s)

'(s)

rr

r

ν

×

=

.

Velocity of ball can be calculated using energy conservation. The system is closed, so

energy must be conserved. Initially the ball is at rest, so at this instant it contains only

potential energy. When it travels along the track, it has potential energy and kinetic energy

(translational and rotational).

22

() 1

(0) (s) ( )

22

zz c

mvs

mgr mgr I s

ω

⋅

= + +⋅

(12)

Where m is the mass of the object, Ic is the moment of inertia about ball’s center of mass

2

1

2

c

I mR=

, rz(s) is the height of the center of mass at position s and b is the height of the

center of mass from the axis connecting the points of contact (Fig.7b).

Using the definition of angular velocity

()vs

b

ω

=

, we can relate it to v(s). Then, above

equation gives ball’s center velocity:

[]

2

2 (0) (s)

()

zz

mg r r

vs I

mb

−

=

+

(13)

Fig. 9a shows ball velocity vector as ball rolls down. It can be seen its increase in intensity,

due to conversion of potential energy. Fig. 10b shows values of velocity during two

revolutions of spiral path. The velocity square is a function of the vertical coordinate for

a given path point rz(s).

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka

7th International Conference

„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021

June 24-27. 2021 Sarajevo, Bosnia and Herzegovina

Fig. 9. Ball velocity a) vector drawn on path b) intensity

Solving equilibrium conditions in normal and binormal directions (eq.11) gives intensity

of path reactions. These are: reaction on outside rail Fo and reaction on inside rail Fi.

Reaction on outside rail increases due to increase of centripetal force. Opposite, reaction

on inner rail decreases and near the end (3/4) of third half-revolution it becomes zero.

Because of this all third part of path was needed to be fenced as was shown on Fig. 2.

Fig. 10. Track reactions on outside and inside rail a) vectors drawn on path b)

intensity

4. CONCLUSION

Rolling ball sculpture gives possibility to apply knowledge that is being acquired during

Mechanical courses on technical faculty. Another aspect is artistic component of design.

Every student can express its vision of sculpture. Sculpture can be conceived as an idea,

modeled in SolidWorks, analyzed and then ‘digitally’ produced by laser cutter. Another

interesting aspect toward finished product is an assemblage. No matter how design is detail

and every aspect is analyzed, there are always some unpredicted circumstances that should

0.1

0.0

0.1

x,m

0.1

0.0

0.1

y,m

0.15

0.10

0.05

0.00

z,m

2

3

4

s

0.2

0.4

0.6

0.8

1.0

1.2

v, m

s

0.1

0.0

0.1

x,m

0.1

0.0

0.1

y,m

0.15

0.10

0.05

0.00

z,m

2

3

4

s

0.02

0.02

0.04

0.06

0.08

0.10

F, N

Fo

Fi

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika

7th International Conference

„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021

June 24-26. 2021 Sarajevo, Bosnia and Herzegovina

be overcome. In this structure, third part of spiral track needed to be fenced, because ball

was jumping out of this part. Analysis was shown that in this part, inner reaction was

become zero due to centripetal force, so ball was in contact only with outer rail of track.

4. REFERENCES

[1] Boes, Eddie. “Rolling Ball Sculptures by Kinetic Artist Eddie Boes.” Eddie's Mind,

www.eddiesmind.com/.

[2] Mini Gear, (2018, July 14). Build Amazing Big Marble Run Machine - DIY.

YouTube. https://www.youtube.com/watch?v=LF0aDmlM5XU&feature=youtu.be

[3] Horibe, T., Mori, K. (2015). In-plane and Out-of-plane Deflection of J-shaped

Beam. Journal of Mechanical Engineering and Automation, 5(1), 14-19.

[4] Dahlberg, T. (2004). Procedure to calculate deflections of curved beams.

International journal of engineering education, 20(3), 503-513.

[5] Žiga, A., Cogo, Z., Kačmarčik, J. (2018). Out-of-plane deflection of J-shaped beam.

Proceedings of the 21th International Research/Expert Conference, TMT 2018.

pp.269-272.

[6] Brezović, M., Jambreković, V., Pervan, S. (2003). Bending properties of carbon

fiber reinforced plywood. Wood research (Bratislava), 48(4), 13-24.

[7] Beer, F. P., Johnston, R., Dewolf, J. & Mazurek, D. (2014). Mechanics of Materials,

7th Edition, McGraw-Hill.

[8] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John

Wiley & Sons.

[9] Wolfram, S. (1999). The MATHEMATICA® book, version 4. Cambridge university

press.

CORRESPONDANCE:

Alma Žiga, Ass. D.Sc. Eng.

University of Zenica

Faculty of Mechanical Engineering

St. Fakultetska 1

72000 Zenica, Bosnia and Herzegovina

E-mail: aziga@mf.unze.ba

Derzija Begic-Hajdarevic, Full professor

University of Sarajevo

Faculty of Mechanical Engineering

Vilsonovo setaliste 9

71000 Sarajevo, Bosnia and Herzegovina

e-mail: begic@mef.unsa.ba

Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka